What connects the idea and the phenomenon world

Imagine a perfect circle, the circle of ideas, that exists only in your head. It already exists perfectly.It exists without drawing a circle with a compass. The circle in the head does not require human action (drawing),It’s already there.
On the other hand, a circle in the real world cannot exist without some action, such as drawing with a compass or approximating it infinitely with a polygon; furthermore, the form is not completed. No matter how accurately you draw a circle with the compass, the line will shift while you are drawing the circle, or the tip of the pen will wear and the line will become thicker.

Originally, a line is “a thing with no width”, so it cannot be seen with the eyes.The visible circle is a circle drawn by something like a line, but wide (this is not a line!). What is the fundamental difference between a perfect circle in the idea world and a circle in the real world?

There are various opinions since the Greek era, but what is the essence of the difference between these two circles? My opinion is it’s all about circles that already exist and circles that can’t exist without any actions. The former has no concept of time, it just exists forever. On the contrary, the latter can only exist in time.Whether the act of drawing a circle or the act of forming a polygon infinitly, any  action can exist only in time.It’s natural, isn’t it?

For example, consider pi.π cannot exist in perfect form in this world. But it’s already perfectly present in Idea. In mathematics, the perfect pi of this idea and the pi constructed in the real world are regarded as the same thing, and are written as follows.

\pi = 3.1415 \cdots

This expression equates eternal existence with eternal action, connecting things that should not be connected in the senses. Then, π is regarded as the limit of infinite actions (algebraic calculation) in the real world. I imagine that (infinite)analysis was born as a demand from the real world, rather than from any other field of mathematics.The differential calculus was born for Newton to describe the law of gravity.Concepts such as differentiation, limit, and infinity may have evolved from connecting the existence of ideas with the actions of the real world.


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